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Beam Deflections and Material Properties Laboratory Exercise Introduction Probably the most common type of structural member is the beam. A beam may be defined as a member whose length is relatively large in comparison with its thickness and depth, and which is loaded with transverse loads that produce significant bending effects as opposed to twisting or axial effects. Whenever a real beam is loaded in any manner, the beam will deform such that an initially straight beam will assume some deformed shape. Remember in Statics we assumed that a beam was rigid, which is not the actual case. In ENGR350 Mechanics of Materials, many relationships for beams will be developed. We will explore several of them. The first is: 1 M (1) EI where is the radius of curvature of the deflected beam at the point of interest. In addition, from your statics background, M is the internal bending moment, E is Young’s modulus, and I is the area moment of inertia of the cross sectional area of the beam about the neutral axis perpendicular to the direction of the applied load on the beam. The area moment of inertia for a beam with a rectangular cross section is given by the expression below: 1 3 I bh (2) 12 where b is the width of the beam and h is the height or thickness of the beam. Objectives The objectives of this experiment are: a) To explore beam deflections which result from externally applied loads. To accomplish this we will measure the Young's modulus of aluminum, brass, and steel. b) To experimentally measure beam deflections and then compare them to analytical predictions. Preparatory Exercise In our January 23 lecture we conducted an experiment with a cantilever beam. The brass beam was 1.5 inches wide, 0.25 inches thick, and 14 inches long. Use data from lecture and equations (2) and (8) from this handout to calculate the Young’s modulus of the beam material. Compare the calculated value to the tabulated value on page 54 of the “Structures” book. Procedure This experiment involves using a simply supported beam loaded as shown in the next figure. This beam is loaded in such a manner that the radius of curvature is constant between the two supports. A mirror mounted to the beam changes angle as the beam is loaded. By reflecting a laser beam off of the mirror and measuring the change in height of where the beam strikes a vertical plane, we can determine the radius of curvature (). Knowing the moment (M) and the moment of inertia (I) we can perform a calculation to find Young's Modulus (E). 1 Part A (Determining Young’s Modulus) The following information should help you in understanding the fundamentals of beam deflection and also allow you to make the proper calculations to determine Young’s Modulus for the various beam materials. d e Mirror Laser Beam Target a c a P P For the above loading condition on the beam, the moment, M, is determined to be: M=Pa (3) From the following figure two more relationships can be found as shown below: e/d = 2 = c/2 (4) d e 2 c/2 Angle of Incidence 2 If the last two relationships are combined the result is: = dc/e (5) Putting all these relationships into the radius of curvature relationship leads to the following: 1 Pa = dc b h3 E e 12 which can be rearranged to give the following: Eb h3 P= e (6) 12dca this has the form of y = mx+b where y = P and x = e. In Part A you will apply a load P and measure an elevation e. You then will do a regression fit to find the slope m from which you can calculate Young's Modulus E. The data that you will need to take is to be found in the Data Sheets. Part B (Comparison of Measured and Predicted Beam Deflections) In this part you will take measurements of beam deflections and compare them to an analytical prediction. The following equation applies to a cantilever beam as shown in the next figure. P x2 v= (3L - x) (7) 6EI where (v) is the vertical deflection of the beam at any point along the length of the beam (x) L x P The Laboratory Assistant will demonstrate the experimental apparatus. See the Data Sheets for the data that needs to be taken. 3 Part C (Deviation of Curvature Equation) One of the main assumptions related to beam deflections is that the deflections are small. From this comes the simplification that when is small, cos 1. However, when beams are exposed to large deflections, this assumption breaks down. This part of the lab addresses the breakdown of the “curvature” equation presented in Part A. For the cantilever beam similar to Part B, the deflection equation for the “end” of the beam is PL3 v (8) 3EI Now apply loads in increments of 10 grams up to 100 grams and measure the deflections. See the Data Sheets for the required data. Laboratory Write-up Follow the laboratory report instructions previously provided. For Part A, do the values for Young's modulus compare well with the values provided in Gordon on page 54? Also find two other sources for Young's modulus. Are various sources consistent? How well should they compare? You measured a, c, d, b, h, e, P during this experiment. An error in which of these would cause the most error in the resulting calculation of Young's Modulus? Why? Draw the general profile of the “Shear and Moment Diagram” for the loading conditions of Part A. Explain why the radius of curvature is constant over the range of “c” from the figure on page 2. For Part B, prepare a plot of P (x-axis) versus v (y-axis) with the analytical expression indicated as a solid line and the experimental data as points. How well do the two compare? What is your definition of well in this case? For Part C, prepare a plot of P versus v with the analytical expression indicated as a solid line and the experimental data as points. How well do the two compare? At what load does it appear the experimental results start to diverge from the analytical expression? If the equation for the slope of the beam at the point of loading is = PL2/3EI, what is the angle at the maximum load applied? How about at the loading where the analytical and experimental results diverge? What is the cos value equal to at the angle you chose? Is it close to 1? Also, in looking at the deflection of the beam at the higher loads, what geometrical changes are occurring to the loading configuration that could influence the deflection results? 4 Data Sheets Part A: Data you will need before beginning the experiment is as follows: Distance from target to mirror (d)=_____________ Distance between supports (c)=________________ Distance from load to support (a)=_____________ Width of beam (b)=____________________ Height of beam (h)=___________________ Initial height mark of laser =___________________ The following table may be useful for taking data. Load P Versus Elevation e Aluminum Steel Brass Load Elevation Load Elevation Load Elevation (lb) (in) (lb) (in) (lb) (in) Part B Before starting this experiment you need to measure the following quantities: The beam width (b) =___________________ The beam height (h) =__________________ The beam length (L+h) =__________________ The distance (x+h) to where 5 you are measuring the deflection =____________________ The initial micrometer reading (Dinitial) =_________________ The following table may be useful for data collection. Load P versus Deflection (v) Load Dfinal v* Load Dfinal v* (lb) (in) (in) (lb) (in) (in) * v=Dinitial-Dfinal Part C: Before starting this experiment you need to measure the following quantities: The beam width (b) =___________________ The beam height (h) =__________________ The beam length (L+h) = The initial deflection of the beam due to its own weight (vinitial) =_________________ 6